Variational Principles for Mirror Descent and Mirror Langevin Dynamics

Belinda TzenAnant RajMaxim RaginskyFrancis Bach

Mar 2023

摘要

Mirror descent, introduced by Nemirovski and Yudin in the 1970s, is aprimal-dual convex optimization method that can be tailored to the geometry ofthe optimization problem at hand through the choice of a strongly convexpotential function. It arises as a basic primitive in a variety ofapplications, including large-scale optimization, machine learning, andcontrol. This paper proposes a variational formulation of mirror descent and ofits stochastic variant, mirror Langevin dynamics. The main idea, inspired bythe classic work of Brezis and Ekeland on variational principles for gradientflows, is to show that mirror descent emerges as a closed-loop solution for acertain optimal control problem, and the Bellman value function is given by theBregman divergence between the initial condition and the global minimizer ofthe objective function.

展开全部

图表提取

论文十问

参考文献

被引用

提问与回答

暂无人提供速读十问回答

被引用

笔记

问答

领域